Moment generating function of a constant

Question

This might be trivial, but can you elaborate why moment generating function for a constant $c$ is $e^{cX}$, where $X$ is a random variable.

Answer 1

It is not. The moment generating function for a random variable $X$ is defined as

$$MGF_X(t) = E\left[e^{Xt}\right]$$

where $t$ is the argument of the function, a real number.

If $X$ is a degenerate random variable, a constant, $X =c$, then plugging this into the above we have

$$MGF_c(t) = E\left[e^{ct}\right] = e^{ct}$$

since the function $e^{ct}$ is deterministic and so equals its own expected value.